Bias Robustness and Efficiency in Model-Based Inference

In model-based inference, the selection of balanced samples has been considered to give protection against misspecification of the model. A recent development in finite population sampling is that balanced samples can be randomly selected. There are several possible strategies that use balanced samples. We give a definition of balanced sample that embodies overbalanced, mean-balanced, and $\pi$-balanced samples, and we derive strategies in order to equalize a $d$-weighted estimator with the best linear unbiased estimator. We show the value of selecting a balanced sample with inclusion probabilities proportional to the standard deviations of the errors with the Horvitz-Thompson estimator. This is a strategy that is design-robust and efficient. We show its superiority compared to other strategies that use balanced samples in the model-based framework. In particular, we show that this strategy is preferable to the use of overbalanced samples in the polynomial model. The problem of bias-robustness is also discussed, and we show how overspecifying the model can protect against misspecification.